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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrssre | Structured version Visualization version GIF version |
Description: A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
xrssre.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
xrssre.2 | ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) |
xrssre.3 | ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) |
Ref | Expression |
---|---|
xrssre | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrssre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
2 | ssxr 11229 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
4 | 3orass 1091 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) |
6 | 5 | orcomd 870 | . 2 ⊢ (𝜑 → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ)) |
7 | xrssre.2 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) | |
8 | xrssre.3 | . . . 4 ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) | |
9 | 7, 8 | jca 513 | . . 3 ⊢ (𝜑 → (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) |
10 | ioran 983 | . . 3 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) | |
11 | 9, 10 | sylibr 233 | . 2 ⊢ (𝜑 → ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
12 | df-or 847 | . . 3 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) ↔ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) | |
13 | 12 | biimpi 215 | . 2 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) |
14 | 6, 11, 13 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 ∈ wcel 2107 ⊆ wss 3911 ℝcr 11055 +∞cpnf 11191 -∞cmnf 11192 ℝ*cxr 11193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 |
This theorem is referenced by: supminfxr2 43790 |
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